WANG Bo,XIE Junwei,ZHANG Jing,and ZHANG Haowei

1.Air and Missile Defense College,Air Force Engineering University,Xi’an 710051,China;2.Highway Railway College,Shaanxi Vocational and Technical College of Transport,Xi’an 710018,China

Abstract: The range and angle information of the frequency diverse array (FDA) cannot be exclusively determined at the output of the array because of the range-angle coupled transmit beampattern.The best decoupling approach is to form a dot-shaped beampattern rather than the S-shaped beampattern of the basic FDA.Considering the degradation of the output signal-to-interferenceplus-noise ratio (SINR) caused by the coupled beampattern, we propose a dot-shaped beamforming method based on the analyzed overlapping subarray-based using a logarithmic offset and a subarray-based planar FDA using a logarithmically increasing frequency offset, with elements transmitting at multiple frequencies. Several simulation results demonstrate the effectiveness of the proposed method in transmit energy focusing, sidelobe suppression and array resolution.

Keywords: subarray-based frequency diverse array (SB FDA),planar frequency diverse array (PFDA), decoupling, dot-shaped beamforming.

1.Introduction

Phased array(PA) radar has been widely used for several decades in civil and military applications to improve signal processing performance.It is one of the most popular types of radar[1–3].However,the range-related interference cannot be directly suppressed by beamforming without multiple antennas or a multibeam antenna based on a PA.In contrast to PA,frequency diverse array(FDA)was initially designed by Antonik to form a time-range-angle dependent transmit beampattern [4–6]. It can achieve a time-range-angle dependent beampattern by adding a relatively small fixed frequency offset between the elements.It has sparked many interesting investigations due to the additional controllable degrees of freedom of the array [7–13].However, the range and angle information of the basic FDA cannot be exclusively determined at the output of the array because of the range-angle coupled transmit beampattern[14–16].The best decoupling approach is to form a dot-shaped beampattern rather than an S-shaped beampattern, which generates maxima at multiple ranges and angle values[17–20].This paper mainly works on the analysis of the dot-shaped beamforming method.

At present, fruitful achievements have been obtained on the dot-shaped beamforming technology. Khan et al.proposed a logarithmic frequency offsets based FDA(log-FDA) to achieve a beampattern with a single maximum at the target location[21].Then some improvements in transmit beampatterns have been shown for multiple input multiple output FDA(MIMO-FDA)by using a variable logarithmic offset, followed by a detailed signal model for better estimation of targets at the receiving side [22–24].However,the FDA using logarithmic frequency offsets achieves a poor resolution in both range and angle dimensions. The pattern synthesis for the FDA radar transmitter is also optimized by jointly considering flexible magnitude response and sidelobe suppression in [25]. Based on the design of the array configuration, Wang proposed a subarray-based FDA radar, which can be used to localize the target in the range-angle domain [26]. The contributions of this paper are as follows. (i) A dot-shaped beamforming method based on the overlapping subarraybased FDA using logarithmically increasing frequency offset (OSB log-FDA) is analyzed. (ii) The mainlobe of the OSB log-FDA is further optimized by considering the array elements transmitting at multiple frequencies.The analyzed dot-shaped beamforming technology is able to offer directional gain to detect weak targets in the target direction and suppress range-dependent interferences from other directions.It can also be widely used in communication,radar,and navigation systems.

2.Data model

2.1 Basic FDA

Fig.1 shows the configuration of the basic FDA[27].Under the narrowband assumption, FDA can be realized by employing a tiny frequency offset relative to the carrier frequency across the array elements. The signal transmitted by thenth element can be written as

wherefn=f0+nΔf(n=0,1,...,N ?1).f0,ΔfandNrepresent the carrier frequency, fixed frequency offset and the number of array elements,respectively.

Fig.1 Configuration of basic FDA

The signal received by a specific far-field target with the angleθand slant rangeRfor the first element is a superposition of the delayed and attenuated version of the transmitted signal:

wherern=R?ndsinθis the range of the target from thenth antenna element,ddenotes the element spacing,andcis the speed of light.

The array factor generated by the FDA at the far-field target(R,θ)can be given as

2.2 Log-FDA

There is a range-angle coupling inherent in the transmit beampattern of the basic FDA because of the linear incremental synchronization between the element spacing and the fixed linear frequency offset.In order to eliminate the synchronization, it is suggested to adopt a nonlinear frequency offset, such as the logarithmical frequency offset,to decouple the beampattern in range and angle domains[21–24].

The frequency fed to thenth element of log-FDA can be rewritten as where lg(·) represents the natural logarithm of a number.δis a small configurable parameter,measured in hertz, to control the frequency offset.

The array factor generated by the log-FDA will be

whererepresents the complex weights, which are used to steer the maximum at the expected far-field target location(R0,θ0).

2.3 Planar FDA(PFDA)

In this paper,we also extend the dot-shaped beamforming method for the basic FDA to planar array geometries proposed in[28],with array elements equidistant distribution in theXaxis andYaxis shown in Fig.2.

Fig.2 Configuration of PFDA

The signal transmitted by the element (n,m) in the PFDA can be expressed as

wherefnm=f0+nΔfx+mΔfy(n= 0,1,...,N ?1,m= 0,1,...,M ?1). Considering a far-field target(x0,y0,z0),R0,?0andθ0of the target can be reached.Taking the element (0,0,0) as the reference of the array,the range between the reference element and the observed point isThen the range between the element(n,m)and the observed far-field target can be expressed asThe signal arriving at the far field target can be written as

The overall signal observed atin the far field can be written as

whereαdenotes the complex weights, which are used to steer the maximum at the expected far-field target location

3.Subarray-based dot-shaped beamforming

In order to decouple the FDA range-angle dependent beampattern, the best way is to form a dot-shaped beampattern without periodicity in the maximum rather than the trailing beampattern of log-FDA.It can be achieved by the design of the array configuration [29–33] as well as the design of the frequency offset[34–36].In this section,we analyze the dot-shaped beamforming method based on the OSB log-FDA.

3.1 OSB FDA

The essence of subarray-based FDA(SB FDA)is to divide the basic FDA into multiple subarrays,which employ the same or distinct frequency offset.Taking the leftmost element as the reference,the configuration of the overlapping subarray-based FDA(OSB FDA)is shown in Fig.3.

Fig.3 Configuration of OSB FDA

The total element number of this array isN. It is observed that the basic FDA is divided intoN ?M+1 overlapping subarrays, respectively, denoted as{S1},{S2},...,{SN?M+1}, each of which consists ofMelements.The signal radiated by the subarrayltoward the specific far-field target(R,θ)can be expressed as

wheresl(t)denotes the transmit signal of thelth subarray,wlis the weight vector,anduldenotes the steering vector:

3.2 Elements transmitting at multiple frequencies

The basic FDA transmits an identical baseband waveform at each element with a coherent single frequency. In this section,we use multi-carrier frequencies on each element of the OSB FDA to decouple the transmit beampattern.

The signal of thenth element simultaneously transmitting thelth frequency component can be written as

wherefnl=f0+Δfn+Δfldenotes the radiation frequency of thenth element transmitting thelth frequency component.Δfn=lg(n+1)Δf(n=0,1,...,N ?1)is the frequency offset between thenth element and the reference element,andNis the array number for the given array aperture.Δfl=lg(l+1)Δf(l=0,1,...,L?1)denotes the frequency offset of each array element transmitting thelth frequency component,andLis the total number of frequency components transmitted by each array element.

The total signal received by the far-field observation target can be expressed as

wherewnldenotes the transmit weight for each array element,and(R,θ)is an arbitrary far field observation point.The transmit beampattern of the log-FDA transmitting at multiple frequencies can be given as

4.OSB PFDA

Most existing reports in the literature of the dot-shaped beamforming only focus on the basic FDA. In practice,the radar application requires the ability to beam steer in higher dimensionalities, which will exceed the capability of the one-dimensional uniform linear array. In this section, we divide the basic PFDA proposed in [28] into the overlapping subarray-based PFDA(OSB PFDA)as shown in Fig.4.

Fig.4 Configuration of OSB PFDA

Fig.4 shows a 6×6 PFDA,which can be divided into four subarray-based FDAs each with 3×3 array elements.The array elements in the green wireframe constitute the subarray 1. The array elements in the orange wireframe constitute the subarray 2. The array elements in the blue wireframe constitute the subarray 3.The array elements in the purple wireframe constitute the subarray 4. The dotshaped beam can be achieved by employing a nonlinear frequency offset across each subarray elements.The transmit beampatterns of the three OSB-PFDAs and the basic PFDA are presented in Figs.11–14 for 10 dB off the peak value.

5.Simulation results

In this section,simulations are performed to verify the effectiveness of the proposed approach.We assume a basic FDA of 20 sensors spaced a half-wavelength apart.The rest of the parameters are listed in Table 1.

Table 1 Simulation parameters

Example 1 Transmit beampattern of the four FDAs.

Fig.5 shows the transmit beampattern of the log-FDA.Fig. 6 shows the transmit beampattern of the OSB-FDA with a fixed frequency offset. Fig. 7 shows the transmit beampattern of the OSB log-FDA.Fig.8 shows the transmit beampattern of the OSB log-FDA transmitting at multiple frequencies.

Fig.5 Transmit beampattern of log-FDA

Fig. 6 Transmit beampattern of OSB-FDA with fixed frequency offset

Fig.7 Transmit beampattern of OSB log-FDA

Fig.8 Transmit beampattern of OSB log-FDA transmitting at multiple frequencies

The decoupling method with a logarithmically increasing frequency offset, which is shown in Fig. 5, achieves a poor resolution in both range and angle dimensions. It can be seen from Fig. 6 that the OSB-FDA with a fixed frequency offset forms a trailing beam in the scanning position. The OSB-FDA with a fixed frequency offset has a certain effect on eliminating the inherent range-angle coupling.However,the multiple maxima at different ranges in Fig.6 are easy to be disturbed.The decoupling method can be optimized by the introduction of the logarithmical frequency offset to remove the periodicity in the maximum as shown in Fig.7.Considering the sidelobe’s magnitude,the performance of the OSB log-FDA is further optimized in Fig.8.

Example 2 Comparison of angle and range dimensional beampattern.

We also examine the resolution in the range and angle dimension of the four analyzed FDAs. Fig. 9 shows the comparison in angle dimensional beampattern.Fig.10 shows the comparison in range dimensional beampattern.

The beampattern of the log-FDA has a wide spread in the maxima, which may result in an ambiguous tracking performance.It is shown in Fig. 9 (b)–Fig. 9(d) that the beamwidth in the angle of these three OSB log-FDAs is between 10?and 20?, while the log-FDA is several times of it.

Fig.9 Comparison in angle dimensional beampattern

Fig.10 Comparison in range dimensional beampattern

It is shown in Fig.10(b)–Fig.10(d)that the beamwidth in the range of the three SB FDAs is also much smaller than that of the log-FDA. We can conclude that the three analyzed OSB-FDA schemes can produce narrow beam steering to the target.With the same total frequency offset,the OSB log-FDA transmitting at multiple frequencies has the best decoupling performance in both angle and range domains.It can also be seen in Fig.11 that the SINR of the proposed SB FDA is better than that of the log-FDA.

Fig.11 Performance comparison of SINR of different FDA radars

Thus, the proposed system has a better robustness against interference.Fig.12 shows the detection probability versus the SNR for the proposed subarray-based FDA and log-FDA.The proposed FDA exhibits better detection performance compared to the log-FDA.

Fig.12 Probability of detection versus SNR

Example 3 Dot-shaped beamforming for the OSB PFDA.

Based on Section 4, we analyze the dot-shaped beamforming for the OSB PFDA in this example. Consider a basic 8×8 PFDA spaced a half-wavelength apart.The target locates at (50 km, 50 km, 150 km), Δfx= 2 kHz,Δfy= 2 kHz, the carrier frequency is 10 GHz. Fig. 13 shows the 10 dB beamwidth of the mainbeam of the basic PFDA with a carrier frequencyfnm=f0+ lg(n+1)Δfx+lg(m+1)Δfy.Then we divide the basic 8×8 PFDA into four 5×5 OSB PFDAs. Fig. 14 shows the 10 dB beamwidth of the mainbeam of the OSB PFDA with a fixed linear frequency offset. Fig. 15 shows the 10 dB beamwidth of the mainbeam of the OSB PFDA with a logarithmical frequency offset. Fig. 16 shows the 10 dB beamwidth of the mainbeam of the OSB PFDA with a logarithmical frequency offset transmitting at multiple frequencies.

Fig. 13 Transmit beampattern of basic PFDA with carrier frequency fnm

Fig. 14 Transmit beampattern OSB PFDA with fixed frequency offset

Fig. 15 Transmit beampattern of OSB PFDA with logarithmical frequency offset

Fig. 16 Transmit beampattern of OSB PFDA with logarithmical frequency offset transmitting at multiple frequencies

It can be seen from Fig. 13 and Fig. 14 that the beampatterns both have a wide spread of the maxima, which may result in an ambiguous tracking performance.Fig.15 and Fig.16 show that the transmit beampatterns of the two OSB PFDAs can both form a dot-shaped beam in the target position.(i)Considering the overlay airspace of the mainlobe,Fig.13>Fig.14>Fig.15>Fig.16.(ii)Considering the sidelobe’s magnitude,Fig.16 is the most ideal.

6.Conclusions

FDA radar has received increasing attention in recent years because of the controllable degrees of freedom of the array.In this paper,the dot-shaped beam based on the OSB FDA is realized. Comparisons of the proposed OSB log-FDA transmitting at multiple frequencies with log-FDA in simulations show the improvement in transmit beampatterns. The subarray-based structure simplifies the processing and assembly of the array,providing a wide signal bandwidth. It also has a wide application prospect in the fields of range-angle joint estimation,front-view detection and imaging of radar targets.Further research will be carried out on the basis of the time-invariant SB FDA.